Optimal Operational Strategies for
an Inspected Component Statement of the Problem
Pulkkinen, U. and Uryasev, S.P.
IIASA Working Paper
WP-90-062
October 1990
Pulkkinen, U. and Uryasev, S.P. (1990) Optimal Operational Strategies for an Inspected Component - Statement of the
Problem. IIASA Working Paper. IIASA, Laxenburg, Austria, WP-90-062 Copyright © 1990 by the author(s).
http://pure.iiasa.ac.at/3393/
Working Papers on work of the International Institute for Applied Systems Analysis receive only limited review. Views or
opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other
organizations supporting the work. All rights reserved. Permission to make digital or hard copies of all or part of this work
for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial
advantage. All copies must bear this notice and the full citation on the first page. For other purposes, to republish, to post on
servers or to redistribute to lists, permission must be sought by contacting [email protected]
Working Paper
Optimal Operational Strategies for
an Inspected Component Statement of the Problem
U.Pulkkinen and S. Uryas'ev
11'P-90-62
October 1990
BIIASA
International Institute for Applied Systems Analysis
A - 2 3 6 1 Laxenburg
Telephone: (0 2 2 36) 715 2 1 * 0 D Telex: 0 7 9 1 3 7 iiasa a
Austria
Telefax: (0 22 3 6 ) 71313
Optimal Operational Strategies for
an Inspected Component Statement of the Problem
U.Pulkkinen and S. Uryas 'ev
1'l7P-90-62
October 1990
WOI-kingPapers are interim reports on work of the International Institute for Applied
Systems Analysis and have received only limited review. Views or opinions expressed
herein do not necessarily represent those of the Institute or of its National hlernber
Organizations.
ElIIASA
International Institute for Applied Systems Analysis
Telephone: (022 36) 715 21 *O
A-2361 Laxenburg
Telex: 079 137 iiasa a
o
Austria
Telefax: (022 36) 71313
Foreword
This is the second report on work done on time dependent probabilities initiated in cooperation
between the International Atomic Energy Agency (IAEA) and IIASA in 1990. T h e treatment of
the underlying mathematical model is rather theoretical, but the intent is t o cover a broad range
of applications. The advantage with the problem formulation is that it enables the inclusion also
of monetary considerations connected t o risks and the actions for decreasing them. T h e intent
in formulating the model is that it will be used for a computerized optimization of selected
decision variables. Originally, the formulation was initiated by the problem of optimization
of test intervals at nuclear power plants. In this paper the non-destructive testing of major
components has been approached. The main result of the paper is the formulation of an optimal
rule for decision if continued operation can be considered safe enough. T h e decision rules
integrates the earlier operational history, safety concerns and economic considerations. Also
other applications are proposed t o be treated within the modeling framework. One specific
problem is the selection of the most suitable time instant for a major repair or retrofittii~gat
a plant. The time horizon of the model can be selected either short-term. stretching only over
a few weeks or long-term, to encompass the complete life time of a depository of spent nuclear
fuel.
Comments 01. proposals for applications of this modeling approach are invited.
Bjorn M'ahlstrom
Leader
Social 8! Environmental Dimeilsions
of Technology Project
Friedrich Schnlidt-Bleek
Lea.der
Technology, Economy a n d
Society Program
Contents
1 Introduction
1
2 General Description of the Model
2
2.1 Description of the Physical Phenomena . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Probability Distribution for the Number and Size of the Defects . . . . . . . . . . 3
2.3 Probability Model for the Shock Occurrence . . . . . . . . . . . . . . . . . . . . . 5
2.4 Model for the Defect Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.5 Probability Model for Inspections . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.6 Probability Model for Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.7 Estimation of the Failure Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3
4
Formulation of the Optimization Problem
3.1 General Descriptioil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Decision Alternatives and Decision Rules . . . . . . . . . . . . . . . . . . . . . .
3.3 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Approximate Calculation of the Objective Function . . . . . . . . . . . . . . . . .
3.5 Statement of the 0ptimiza.tion Problem . . . . . . . . . . . . . . . . . . . . . . .
12
Conclusions
19
12
13
15
18
18
OPTIMAL OPERATIONAL
STRATEGIES FOR AN
INSPECTED COMPONENT STATEMENT OF THE
PROBLEM
U. Pulkkinen and S. Uryas'ev
1
Introduction
The failures of mechanical components often develop gradually. This fact should be taken into
account, if we would like to predict failures and optimize operational procedures and strategies
for preventive maintenance and repair of the system. The prediction of failures is usually based
on probabilistic models, the simplest ones being the probability distributions for the time to
fail with deterministic failure or hazard rate. More general models which may describe also the
gradual development of the failures are based on the notion of stochastic intensity, which has
been discussed by some authors (see. for example. [11].[15]). Mathematically, the models \\pith
stochastic intensities are much more difficult than the traditional models.
In many practical cases the gradual development of the failures cannot be observed directly
but through more or less imperfect measurements or inspections. The models for imperfect
observations are also probabilistic, and this leads t o filtering problems in which the stochastic
intensity is estimated on the basis of imperfect observations. This problem has been intensively
considered in the literature (see, for example, (71,[10'1,[13]).
The optimization of the repair and maintenance strategies is usually made by minimization of the expected cost due t o the failures and the maintenance and repairs. T h e practical
means for controlling failure behavior are often very limited, the only possibilities are preventive
maintenance or replacement of the component, and sometimes stopping the operation of the
component. Since the controlling actions are made on the basis of imperfect measurements, it is
also desirable t o optimize the inspection or testing policies. In practice this means, for example.
the selection of inspection intervals, methods and strategies.
Mathematically, the optimization of strategies for inspection, repair and preventive maintenance can be formulated as a stochastic optimization problem. As a rule, analytical solutions
for such problems do not exist, and some numerical algorithms should be used. In principle,
many problems of this type can be solved by using the dynamic programming algorithm (see [2]
and others). Usually the dimension of the problem is very high and in practice one has to apply other approaches, for example the stocllastic quasi-gradient techniques (see [4],[8], [12] and
others) or scenario analysis (see [14]). We are going t o use stochastic quasi-gradient algorithms
with adaptive parameters control t o solve the problem.
The models or approaches outlined above are useful in making decisions concerning the safety
and economical operation of nuclear power plants. This is due t o the stochastic nature of the
failure phenomena and the high cost caused by accidents and extensive inspections of the plants.
The failures of nuclear power plant components which develop gradually are numerous. The
growth of defects in the pressure vessel and in the pipings are good examples. A similar type of
phenomena occur also in the pipings of the steam generators at PWR-plants. The inspections
of the defects are usually very expensive, and the inspection costs depend on the ability of the
inspection method t o reveal the defects. The information obtained from the inspections is used
in the decisions, for example, for stopping the plant or for preventive maintenance, which may
lead t o rather large costs. The problem is t o make optimal decisions in order t o gain economical
profits and maintain a sufficient safety level of the plant.
This paper describes a model t o evaluate and predict the gradual development of failures
of a system which is inspected periodically. Further, an optimization problem is formulated in
order t o find the most appropriate inspection and maintenance policies.
General Description of the Model
2
2.1
Description of the Physical Phenomena
The manufacturing process of any mechanical component is more or less imperfect. IV11e11 the
component is taken into operation it has always some defect or faults, which do not cause the
failure of the component or the system immediately. The number, the size and the type of
the existing defects are usually unknown because they cannot be observed directly. The only
possibility t o evaluate these is t o make good guesses on the basis of experiences obtained from
similar manufacturing processes and components. The uncertainty about the defects can be
modeled with suitable probability distributions.
T h e characteristics of each defect (e.g., size, shape) will change with time. T h e rate of growth
of the defect may be dependent on several environmental conditions, and the number of defects
may increase. Often the development of the defects is correlated with the shocks wllicll occur.
in the system due t o some external phenomena. For example, a t a nuclear power plant typical
shocks may be thermal transients due t o spurious or inadvertent operatioil of safety systems or
due t o emergency shut down of the reactor. The empirical and theoretical research carried out
on this subject is wide (see, for example, [3],[5],[6]).
The development of the defects is followed by making inspections periodically and possibly
a t any moment of time when there is rea.son t o believe that the defects have increased. T h e
probability of identifying and properly estimating cha.ra.cteristicsof the defects depends on the
properties of the inspection methods. In the literature, there are some rather reliable models for
describing the effectivity of inspections, and a lot of experimental research has been made on
the subject ( [1],[9]). In the case of metallic components, the most popular inspection methods
are based on radiographic or ultrasonic inspections or eddy current measurements.
The component fails if the size of defect exceeds some limit. In the case where several defects
exist in the component, it is not easy to express the exact failure criteria. The component may fail
even if all the sizes of the defects are under the failure limit. It is feasible t o think that the failure
probability can be expressed as a function of the failure rate or intensity. In our case, the failure
intensit'y depends on the number and cha.racteristics of the defects in the component. Since
the defects may grow or change in time stocl~astically,the failure intensity is also a stocha,stic
process.
In the following sections we will describe a probabilistic model for the phenomena, described
above. The model gives an idealized picture of the real degradation process of a (mecha,nical)
component, and can be used t o find optimal inspection a.nd repa.ir strategies.
Probability Distribution for the Number and Size of the Defects
2.2
The initial defects in a metallic structure may be classified according t o their properties and
growth mechanisms. The most important properties of a defect are its size and its orientation,
which determine the probability with which they can be identified in an inspection. In principle.
any defect can be characterized with some vector
vector
2
2
= (tl,.. . , z,), where each component of the
corresponds to some property of the defect. This kind of characterization would lead
t o more complicated models than what is needed in our example.
We assume that a defect is completely characterized by its class, denoted by D with D E
D
( 1 , . . . ,Ii),and by its size, denoted by C with C E (O,C:,,].
Interval (O,C,,,]
D
includes C,,,
and does not include 0. The class of the defect is needed because it is possible that all defects
do not have similar growth mechanisms, or they cannot be identified with the same probability.
We suppose that a t the beginning of the operation of the system ( a t time point t = 0) the
initial number of defects is M ( 0 ) 2 0. Each of the h!(O)
defects are characterized with tlieir
class, D,(O), and their size, C,(O), i = 1 , . . . , M(O), at t = 0.
The values of M ( 0 ) . D(0) = ( D ( o ) , . . ., D ( 0 ),,)
and C(0) = ( ~ ( 0 )... . ,C(O),,,,,)
are
unknown, a.nd they depend on the random variation of the manufacturing process. Thus it is
possible to assume tha.t they are random varia.bles specified on the probability space ( P , F ,0)
'.
The model for h4(O), D ( 0 ) and C ( 0 ) is now simply their joint distribution:
P(M(O) = m , D(O) = dl C(O) E [c, c
+ dc)) =
def
MDC
Dm(0) = dm,Cm(O) E [cm,cm+ d c m ) ) = go
(1)
(m,dl,cl,...,dm,cm)dcl x . . . x d c m .
It should be noticed that the random variables M(0) and D ( 0 ) are discrete and the random
,
. . .,(Dm(0), ~ ~ ( 0are
) )indepenvariable C(0) is continuous. If we assume that ( D ~ ( O )Cl(0)),
dent given M ( 0 ) = m, and if the joint distribution of (D,(o), c,(o)) is g t c ( d i , ci)
, then
the
distribution (1) can be written in the form:
MDC
90
M
m
DC
( m , d l , c l , . . . . d n 1 7 c m ) = g o ( n l ) n g o( d i , c i ) ,
i=l
in which g t ( n z ) is the distribution of hd(0).
If we finally assume that there is only one class of defects we obtain the distribution
I11 practical situatioils the assumption of independency of the defect sizes may not be a.ppropria.te, a.nd in that ca.se we cannot write the above distributions in the product form. This may
cause some calcula.tional difficulties.
The functiona.1 form of the distributio~l( 3 ) can be selected from rather wide family of distriM
butions. In our ca.se we restrict the distribution go ( m ) on a set m E (1, . . . ,M,,,)
being rather small (A&,,
with Al,,,
Z 50). Further, it is probable that the size of any defect is small in
the beginning of the operation and t11a.t its size doesn't exceed some upper value. Here we use
the following discrete distribution for the number of the defects:
with
vm 2 0 and
C Z ; '=~1. ~
Here we assume that the size of a defect will follow a truncated exponential distribution with
the density function:
in which 0
< c 5 c,,,,
oc is a known parameter, and Qc is a normalizing factor.
'We denote the random variables without the index w
by z , whenever it is possible without confusion.
E R, i.e. the random variable
z ( w ) is denoted simply
2.3 Probability Model for the Shock Occurrence
The defects of the component may grow due to the shocks which occur in the system. These
shocks are caused by various external phenomena independently of the development of the
defects. These kind of phenomena are usually described with random point process models.
The most simple point process is the time homogeneous Poisson process which we shall apply
here.
We assume that the shocks occur according to the homogeneous Poisson process model with
constant intensity y. Accordingly, the time points a t which the shocks occur,
TI,
I = 1,2,.
. .,
can be expressed as sums of exponentially distributed random variables, i.e.,
in which the varia.bles bl, . . .,6[ are identically exponentially distributed random variables with
density function
g6(t) = y esp {-yt)
.
(7)
Generalizations of the above model can be easily developed, for example, the intensity y may
be assumed time dependent, or even stochastic and dependent on the other random variables of
the model.
2.4
Model for the Defect Growth
In order t o describe the random changes of the number and the sizes of the defects, we have to
make some assumptions. First, we assume that the number of defects will be constant if tlie
component is not repaired. Further, we a.ssume that the size of the defects may increa.se only
a.t the moments of the time where shocks occur (at the time points r l ) . If the component is
repaired, all the old defects will be removed but some new defects may be introduced into the
component. The repair may only be done after some shock or inspection point according to the
control law which will be described later.
The above assumptions are not the only possible ones. For example, the defects may grow
also between shocks due to some chemical phenomena. Further, it is possible that new defects
may be introduced into the component also between repairs. Thus our assumptions must be
considered as idealized approximations.
We denote the number of defects a.t time point t by M ( t ) and the vector of sizes of the defects
a t the same time point by C(t). The time points where the inspections are made are denoted by
t i , t i , . . . We denote the ordered union of the points t:, t i , . . . and T I ,TI,. . . by V = {tl , t2, . . .).
Figure 1: Set V = { t l , t 2 , . . .)
At each 1, there either occurs a shock or an inspection (and possibly a repair) is made. T h e set
\/ is illustrated in Figure 1.
Due t o our assumptions, the changes of M ( i ) and C ( t )must be considered only a t the time
points i,. A l ( i ) and C ( i )are constant in any interval [i,,t,+l), i.e.
and
If t h e component is repaired a t the time point t j then the old defects are removed from the
component. However, some new defects are introduced into t h e component. T h e number and
the sizes of these new defects follow the distributions
Afm,,
with
r);
>_ 0 ,
r);
= 1, and
m=l
where 0
5 c 5 cLax and uRc is a parameter, and QRc is a normalizing factor.
T h e values of the process A/l(t) are constant between repairs, or
in which the time point tj' is the point where the component is repaired, and M ( t j ) follows the
distribution (10).
The values of the process C ( i ) are coilstant between the shock points
71,
but they may grow
a t each shock point. The increase of the size of the defect is a random variable which may depend
on the size of the defect just before shock. We assume also that the defects grow independently.
The conditioilal distribution of the size increase of a defect given the defect size, c, is assumed
t o be of the form,
which means that the size increments are exponentially distributed random variables with the
expected value
t.
The process C(t) is constant between shocks (if there are no repairs between the shocks) or
between successive repair and shock:
C ( t ) = C(max {mas rr , mast;)) ,
TlLt
t;<t
in which each conlponent of the vector C ( 0 ) follows the distribution (5), components of C ( t i )
follow the distribution (11)and a t 1 =
T,
the increments of C ( t ) components are random variables
followi~lgthe distribution (14). The increase of the size of one defect is illustrated in Figure 2.
The above model is one of the most simple possibilities. It is applied here in order t o
formulate the optimization problem. In literature (see [3],[6])several different models have been
considered. The model discussed here may be modified rather easily in order to make it more
realistic.
2.5
Probability Model for Inspectioils
The i~lspectionsare made in order t o mea,sure the sizes of defects. However, the inspections
are not complete and they will not identify all defects with probability one. Further, the measurements do not give exact information on the size of the defects. Berens (see [I]) discusses
both the so called hit/miss model and the signal response model for modeling the reliability of
inspection of metallic structures. We shall apply here the model based on the signal response
approach.
The basic idea of the signal respoilse model is that each defect causes a signal which can be
measured. However, there is also measurement noise due t o some external phenomena. Further,
if the signal is too weak then the defect cannot be identified. We denote by t9,(tjj the signal
caused by defect i with the size C i ( t j ) a t the time point t = tj. If the signal t9;(tj) is below some
limit, dtr, then the defect i is not identified. We assume tha.t the signal di(tj) is related with the
true size C,(tj ) according t o
Figure 2: Graph C;(t ).
in which Po and
PI
are paramet,ers and
mean and variance a? (i.e. (
111
N
< is a normally distributed random variable with zero
N ( 0 , a ; ) ). In other words, t h e conditional distribution of
di(tj) given Ci(tj) is normal with parameters ,do + ,dl In Ci(t,) and at. T h e random variable
<
describing tlie mea.surement noise could also follow any other distribution, but we use here the
Gaussian distribut.ion.
We assume further t h a t t h e defects are inspected independently, and thus t h e signal di(tj)
corresponding t o tlie defect i is stocl~asticallyindependent of the other signals. T h e probability
t h a t a. defect of t h e size c is not identified in t h e inspection is given by
in which
@ { a }
is the cumulative standard normal distribution function.
Let us denote by r/(tj) t h e number of defects identified in an inspection a t t = t j . Since
all defects are not identified with probability one, v ( t j ) may be smaller than the true number
of defects M ( t j ) , i.e. v ( t j )
5 M ( t j ) . T h e result of an inspection a t t = t j is described with a
random variable ( v ( t j ) , d Y ( t j ) )= ( ~ ( t j )dl(tj),
,
dz(tj), . . . ,d Y ( t , ) ( t j ) ) . ~ e f i n e
T h e conditional probability distribution of ( v ( t j ) , dl(tj), d2(tj), . . . , dY(tl)(tj))given
( ~ ( ~ j ) ? ~ l ( ~ ' .~' 7)c M
~ (~t l )2( t(j )~)is j ) ~
in which
IO .
otherwise ;
is the truncated density function of normal distribution with the parameters Po
and a; , and normalizing constant Qt'
.
+ P1In Ci(tj)
The distribution (19) has the above form due to
the independence of di(tj) given ( ~ ( t j )~,( t j ) ) It
. should be noticed that in (19) the defects
are indexed in the order of identification (the defects 1 , . . . ,v(tj) are identified, the defects
u(tj)
+ 1,. . . ,h I ( t j ) are still latent after the inspection).
The above model describes a situation in which only one inspection is made. In pra.ctice,
the componeilt is inspected periodically and possibly by applying several inspection methods.
In order to describe this situation we have to extend the above model. First we assume
tl1a.t each inspection method can be described with the model given in the equations (17), (18).
The only difference between the inspection methods are the parameters values of the respective
normal distribution (the parameters Po, P I , a:).
In the following we need these parameters for
two inspectioil methods and we denote them by Po,, PI,, a:,
with k = 1,2.
The simplest way to model a series of successive illspectioils is to assume t11a.t the successive
inspections are stocl~asticallyindependent. This assumption ha.s some practically unacceptable
consequences. For instance, it is possible that a defect which has already been identified in earlier
inspections will not be identified again. In practice, the known defects are usually identified at
every inspection after the first identification. This means that the successive inspections are
dependent which should be taken into account in the model. Here we apply the following simple
approach. We assume tha.t the result of a.n inspection depends on the result of the previous
inspection such that if the defect was identified at the previous inspection (i.e. di(tj-l)
> 0''
for the defect i) then the result of the present inspection would also exceed the identification
threshold Ot'.
Further, we assume that the conditional distribution of the result (u(tj), ou(tj))
of the present inspection j , given tha.t the defect was identified at the previous inspection j - 1
is the truncated normal distribution. More exa.ctly
In the above equation (21) the first product describes the defects identified a t the previous
inspection a t t,-l
and a t the present inspection a t t j and the second product describes the
defects which are still unidentified.
If there are several types of inspections we assume that each one follows the above model
(with their own parameters), and that if a defect is identified at a previous inspection of any
type then it will be identified a t every future inspection of any type.
The above model for inspections is rather simple. In a real situation it is possible that the
results of future inspections will be dependent also on the value of the previous inspections (not
only on the fact that the defect wa.s identified as in our model). In some cases the inspections
may be so noisy tha.t they indicate the existence of a defect although there is no defect.
2.6
Probability Model for Failures
A component with ma.ny large defects will obviously fail with higher probability than a component without any defects. However, also a component without any defects from the described
class will fa.il after a. rather long period of time. Here we assume that the increase of the failure
intensity due to a. defect will be proportional to the size of the defect. Thus the failure intensity
of the component can be written in the form
where A1(t) is the deterministic pa.rt of the failure intensity and the sum describes the stocha.stic
contribution due to the defects.
In the above equation h4(1) (the number of the defects a t time t ) develops stochastically
according t o the model given in the equations (8,10,12,13) and C ( t ) (the sizes of the defects)
behaves a.ccording t o the equations (9,11,14,15,16). The development of the variables M ( t ) and
C ( t ) depend also on the occurrence of shocks and the decisions made t o control the failure
intensity.
The deterministic pa.rt of the failure intensity, A1(t), is a.ssumed to be of the form
in which
a0
and
a1
are nonnegative constants. Figure 3 describes the process A(t). The increase
.
of the failure intensity, AA(r,) in Figure 3 is given by
Figure 3: Graph X(t).
The conditional survival function (i.e. the probability that the component will survive over
the time period [O:tj]) is (see, for example [15])
def
S(tj) = P(T, >
tj
I X(t),O 5 t < t j )
= exp
in which X(t) is defined in the equation (22) and T, is the failure time.
2.7
Estimation of the Failure Intensity
The inspections ma.de in time points tJ give information on the number and the size of the
defects in the component. Since the failure intensity is related to the size and the number of
the defects this information can be used in the estimation of failure intensity at each inspection
point.
In fact, the joint conditional distribution of the variables M ( t ) and C ( t ) given the results
of inspections until t (i.e. (u(o), O(0)), . . . , ( u ( t j ) , O(tj)), t j 5 t < t j t l ) and the time points
( r l , . . . ,TI,TI 5 t <
TI+~
)
where shocks have occurred, can be determined recursively by using
the law of the of the defect size increase, the law of the shock occurrence and Bayes rule. By
using this conditional distribution we could also determine the respective conditional failure
proba.bility. In this study we do not consider these distributions, but we 'estimate' the failure
intensity directly using the information from inspections.
Let us assume tha.t t h e information up to the time t is the following set:
in which t j
5 t < t j + l , rl 5 t < rj+l. T h e estimate of the failure intensity is a function of the
variable listed in the above set, for instance of the form:
in which A(.) is a function satisfying some measurability requirements.
Here we take into account only t h e result of the most recent inspection, i.e. (u(tj), eu(tj)).
Since the result of the inspection and t h e variables M ( t ) , C ( t ) a r e related in a very simple way
(see the equation (IT)) and the failure intensity is a simple function of M ( t ) and C ( t ) (see the
equa.tions (22), ( 2 3 ) ) , we have the following estimate
where p and X1(t) are defined in the equations (22), (23) and
Po and
/31 a r e defined in the
equation (17).
Formulation of the Optimization Problem
3
3.1
General Description
T h e cornponeilt described above is used t o give profit t o the operator. Thus it is preferable t o
use it a.s long as possible. If the system is used over a long time, then the probability of failure
increa.ses. T h e losses due t o the failure ma.y be very high, especially in the case of nuclear power
plants. T h e operator or the decision maker ( D M ) has the possibility t o avoid high costs due t o
the failure by stopping the opera.tion of t h e component or by repairing it. T h e early stopping
will lead t o loss of profit and the repair may be very expensive.
T h e decisions either t o stop the component or t o repair it may be done on the basis of the
information obtained from t h e inspections. However, the inspections a r e also expensive and they
cannot be made very often. T h e DM ha.s t o decide how often a n inspection should be made and
which type of inspection should be used.
Since we have described the stocha.stic behavior of the component we may formulate the
selection of the best decision as a stochastic optimization problem. For that purpose we have
t o determine t h e objective function, give the model for the dynamics of the component, and
establish the decision rules.
Figure 4: Shifted inspections schedule.
3.2 Decision Alternatives and Decision Rules
We assume that there are two types of inspections which follow the models described in Section
2.5. The most extensive inspection, 'the large inspection' is made regularly or periodically with
fixed interval T I . The cost of the large inspection is G I . The other type of inspection, 'the small
inspection', is made after every shock. The cost of the small inspection is Gz. Depending on the
result of the small inspection also a large illspectioil is made after shock, in that case the time
sclledule of the large iilspections is shifted. This means that if the shock occurred a t the time
point
rl
and the decision was t o make also a large inspection a t
inspection will be made a t t = rl
rl,
then the next regular large
+ Tl (see Figure 4).
After obtaining the result of a large inspection a t some time point t, E I f (see Chapter 2.4
for definition of V ) . the Dhl has to choose between the following alternatives:
continue the operation of the component;
repa.ir the component;
stop the operation of the component.
The decision after a large inspection is denoted by ul ( t j ) with the following values:
u1(tj) =
I
0,
continue the operation without repair ;
1,
repair the component ;
2,
stop the operation of the component.
(27)
After obtaining the result of a small inspection, the DM has to choose between alternatives:
continue the operation of the component;
make a, large test a.nd shift the time schedule.
M'e denote the decisioil a.fter a small inspection by u2(t3) which has the following values:
t
I
U2(tj)=
0
i
.
1,
continue the ooera.tion :
make a large inspection, shift the time schedule.
The selection between the above alterna.tives a t each time point t j should be based on the
information obtained until the time point tj. Further, the decisions should be such that they
minimize the expected value of total costs.
We assume here that the decision is made on the basis of the most recent failure intensity
estimate described in the equation (26). At the time point tj, the DM measures (v(tj), eu(tj))
and uses this value in selecting the best decision. In order t o be able t o find the best decision,
the DM has t o follow some decision rules which are of fixed form.
Here we consider only one class of decision rules. In principle, any other rule could be chosen,
and possibly they would lead t o better solutions (to better value of the objective function). We
assume t11a.t the decisions are ma.de on the basis of the following rules:
o,
if
q l ( i ( ~ , ) , z ~ ( t<~ o) ),
1,
if
q l ( i [ t , ) , ~ l I ( ~ , )2) O , 9 2 ( i ( t j ) , v 2 [ t j ) ) > O
2,
otherwise ,
,
(29)
and
where
91 :R2 + R , 9 2 :R 2
-R ,
q 3 : R 2 -- R are monotone and continuous functions with respect
t o both va.riables. and vl : R - R , v2 :R -- R , v3: R + R are monotone and continuous functions.
The interpretation for the above decision rules is rather simple: the decisions are made if the
recent failure intensity estimate exceeds some thresholds. The sense of the functions q1,
92.
q3
can be explain as follows:
if q l ( i ( t j ) , v l ( t j ) ) < 0 , then continue the operation of the component;
.
if q 2 ( i ( t j ) , v2(tj)) > 0 then make a repair of the component (in case of
~ l ( i ( t , ) , ~ l ( t j2
) )0 ) ;
if 503(;\(tj), v3(tj)) < 0
then continue operation after small inspection without large
inspection.
We shall coilsider only the following forms for the functions q l , 92, q3:
Functions v l(I), v2(t), v3(t) are the controls in the model. Here we consider that they are
linear:
Define x = (xi, x:, x:, xi, x;, x;)
. Now the optimization problem
is t o find the vector x which
leads to the smallest expected costs. In more general cases the respective problem could consist
of finding the optimal forms of the functions
(p,
v,, p = 1,2,3, and of finding the optimal
failure intensity estimate.
3.3
Objective Function
The objective of the decisions described above is t o minimize the costs due t o the use of the
component. The total costs depend on the stochastic development of the defects in the component, and on the choice of the threshold fuilctio~ls~ ~ ( 1 , ) . The costs will be very high, if the
component has failed, and the profits gained from the use of the component will be larger if the
component is used over a longer time. The total cost will be different for each trajectory of the
stochastic process consisting of the values of the variables A!(t),
C(t),
v(t), QV(t), etc.
We denote the profit of the operation of the component per unit time by
of failure by
G, and the cost
Gj.The cost of repair is denoted by G,. The operation of the component mill
be terminated at the time point
T,,,, if it is not terminated earlier (the time interval [0, T,,,]
is the time horizon of the optimization problem). We designate by
Tstop
the stop time for the
component due t o decision rule ~ ~ ( 1 =
, )2 :
Denote by
where
Tend
the termination point
T, is the failure time of the component. The termina.tion point is a random variable
depending on control vector x and the random state of the nature, w E 0,i.e.
Tend
= Tend(x,
w).
Let the number of the large inspections (the small inspections) during the time interval
[0, Tend]
be N1( N 2 ,
respectively) and let the number of repairs during the same interval be AT, .
Now the total cost is
f(x,w)=
where
G(x,w ) ,
Gj+ G(x,w ),
if the component ha.s not failed ;
otherwise ,
We consider t11a.t the objective functio~lF ( x ) is the expected value of the cost function f ( x ,w )
and f ( x , w ) is defined in the equation ( 3 3 ) .
The objective function (34) is not easily evalua.ted. The expected value cannot be determined
analytically due t o the complicated structure of the stochastic processes involved. However, the
expected value can be evaluated in the form
in which the first expectation is evaluated with respect t o a-algebra generated by the the
processes X ( t ) , i ( t ) , 0
5 t 5 T,,,
. It
should be noticed that time points t j and decisions
The conditional expectation
u1( t j ),u 2 ( t j ) are random varia.bles measurable with respect t o FA.
in the formula. ( 3 5 ) may be calculated analytically.
The costs due t o iilspections or repairs may increase only a t the time points t = t j , where
the decisions are made. The failure intensity develops independently on the decisions between
the time points where repa.irs are ma.de. At repair points the failure intensity changes, and
if the decision a.t some time point is t o stop the operation of the component, then the failure
intensity will be equal t o zero. Thus the fa.ilure intensity is a stochastic process which depends
on the decisions made at time points i j in a simple way. Given the trajectories of the processes
X(t), i ( t ) ,0 5 t
< Tstop , the conditioilal expectation failure probability in the time interval
[0, TstOp]is calculated a.s follows (see ( 2 4 ) )
in which the failure intensity X ( t ) is a random function defined in the equation ( 2 6 ) . Consequently the conditional expectation of the failure cost is equal to
The expected costs given X ( t ) , i ( t ) , 0
5 t 5 Tstop, due t o the inspections and repairs can
be written in analogues form. Define jstOpas follows
The costs due t o the large and small inspections are given by
fat-
and
G/(I- S ( T ~ ~ , ,.) )
where
( t j ) , X 2 ( i J )are random variables depending on the failure intensity estimates and they
are defined by the equations
and
T h e respective expected cost due t o repairs are given by
j=1
where x T ( t j ) is a, random variable defined with
T h e random variables ~ 1~ ,2 y,, in the above equations are indicators, which determine the
control action (continue the operation, make an inspection ets.) according t o the rules given
in equations (29),(30). These random variables depend on the vector x , which is t o be chosen
optimally.
T h e respective conditional expectation must be determined also for -GpTend, the expected
profit due t o the operation of the component. The conditional expectation is of the form:
in which
is the conditional expectation of the failure time, given t h a t the component failed during interval
[O,Tstopl .
By collecting the above formulae we obtain the conditional expectation of the cost function
f ( x , w ) given X(i), i ( t ) , 0
< t 5 T,,
in the form:
Jstop
(
1
-t
o
p
)
+ C { s ( l j ) [ ~ l \ l ( i j )+ G 2 ~ 2 ( t j+) ~ r ~ r ( i j ) ] } -
From the above equation and (35) we finally obtain the objective function.
Approximate Calculation of the Objective Function
3.4
We consider that in the model described above, the fa.ult probability of the component is small
value and
Consequently
(- 1
Tatop
s ( t j ) = exp
*(t) dt} z 1 ,
Substitutioil of the ( 4 3 - 45) in the expression (41) gives
def
E [ f ( x , w ) 1 3 x 1 z f(x,w) =
Further we use functioll f ( x . w) to formulate stochastic optimization problem with respect to
vector x .
Statenlent of the Optiillization Problem
3.5
The stochastic behavior of the colnponent (see Section 2), t,he possible decisions and the corresponding decision rules (see Sectio~l3.2) and the objective function are now defined. Let us
designa.te by X a fea.sible set for decision vector x
here 21
21, 1 = 1 , . . ., 6 are low a.nd upper bounds for variables 21, 1 = 1 , . . . , 6 . Now we are
ready to state the optimiza.tioi1 problem. It can be given in the standard form
F ( x ) = ~ [ ~ [ j ( x , w ) I 3 ~ ] ] min,
+
xEX
subject to the dynamics of the process X(t) ,i ( t ) and the decision rules. T h e dynamics of these
processes are described in Sections 2 and 3.2.
4
Conclusions
The aim of this paper is t o formulate an optimization model for finding good operational strategies for an inspected component. In the formulation of the model, the costs of inspections,
repairs and failures and the profit earned from using the component were taken into account.
The purpose of the optimal operational strategy was assumed t o be the minimization of the
costs accumulated during the operation of the component.
We suppose that the component fails randomly and that the failure intensity of the component depends on the number of initial defects in the component. Further, we assume that
the defects grow stochastically. The models for the stochastic failure intensity and the defect
growth are rather simple, they would need further development before they are used for practical
applications.
The model for the illspection of the component was adopted from the theory developed for
evaluating the reliability of non-destructive testing of metallic structures. The inspections are
not complete, and therefore the model was probabilistic.
The above mentioned elemelits of the problem led to a very complicated (from a numerical
point of view) stochastic optimization problem. Although the models applied to describing of
the phenonleila were simple, the optimization problem appears to have several difficult and
interesting features. In this preliminary paper, we did not try t o solve all this problems.
The analytical solution of the problem is not possible, since the objective function is a
very complicated integral (mathematical expectation). For this reason we are going t o apply
stochastic quasi-gradient algorithms. Due to the nature of the algorithm, it is useful t o express
the objective function as a double expected value; first we evaluate the conditional expectation
of some function and finally calculate the unconditional expectation. The solution techniques
will be discussed in future papers on this subject.
The optimal operational strategies described in this paper are the optimal stopping and repair
time of the component and the inspection strategies. The interesting problem connected with
the inspection strategy is t o find conditions for gathering more information on the latent defects
with the component. Principally, this problem may be solved by using the model described
above.
The control rules applied here are some sort of threshold control rules. The actions were
assumed to be made on the basis of the failure intensity estimate, evaluated from the results of
inspections. In our opinion, this kind of rules may be easily implemented in practical situations.
The optimal values of the threshold parameters depend upon the cost structure and on the
parameters of the failure models.
Practical applications of the model are possible if there exist reliable data for the parameters
used. I11 this respect the most problematic part of the model is the probability model for the
defects growth and failure intensity. Some da.ta are available, but it is not evident that they are
enough t o estimate parameters of the model with sufficient accuracy. The d a t a for the inspection
model seem t o be rather reliable.
The model discussed here is designed mainly for applications in nuclear power plant safety.
However, there are a lot of other applications which can be even more fruitful. One possjble
area for this kind of model is the condition monitoring of mechanical components.
References
[I] Berens, A.P. (1976): NDE Relia.bility Data Analysis. In Metals Handbook. pp. 689-701.
[2] Bertsekas, D.P. (1976): Dynamic Programming and Stochastic Control. Academic Press,
New York. 397 p.
[3] Bogdanoff, J.,L., Kosin, F., (1985): Probabilistic Models of Cumulative Damage. John Wiley
& Sons, New York.
[4] Ermoliev, Yu. (1983): Stochastic Quasi-Gradient Methods and Their Applications t o System
Optimization. Stochastic, 4. pp. 1-36.
[5] Gheorghe A., (1982): Applied Systems g! Engineering. Editura Academiei, Bucuregti; John
Wiley & Sons, Chichester, New York, Brisbane, Toronto, Singapore. 342 p.
[GI Iiosin, F., Bogdanoff, J.,L, (1989): Probabilistic hIodels of Fatigue Cmck Growth: results and
speculations. Nuclear Engineering and Design 115, North-Holland, Amsterdam. pp 143-171.
[7] Liptser, R., Shiryaev, A. (1976): Stahstics of Random Processes. Vol.11, Applications.
Springer, Heidelberg. 339 p.
[8] Ljung, L. and T . Soderstrom (1983): Tlieory and Practice of Recursive Identificatioiz. h4IT
Press. 529 p.
(91 A summary of the PISC-Ilproject. PISC II report No.1
-
September 1986. Commissio~lof
the European Conlmu~litiesGeneral Directorate XII, Joint Research Centre, Ispra Establishment and Organization for Economic Co-operation and Development, Nuclear Energy
Agency, Committee on Safety of Nuclear Installations.
(101 Snyder, D.L. (1972): Filtering and Detection for Doubly Stochastic Poisson Processes. IEEE
Tmns. on Inform. Theory, vol 18, no.1, pp. 91-102.
[ l l ] Snyder, D.L. (1976): Random Point Processes, John Wiley & Sons, New York, London,
Sydney, Toronto, 48.5 p.
[12] Uryas'ev, S.P. (1990): Adaptive Algorithms of Stochastic Optimization and Game Theory,
Nauka, Moskow, 173p.
[13] Van Schuppen, J . H . (1977): Filtering, Prediction and Smoothing for Counting Process Ob-
serva.tions, a Ma.rtinga1e Approach. .91.4JI J. Appl. Math., vol 32, no.3, pp. 552-570.
[14] Wets, R.J.-B. (1988): The aggregation principle in scenario analysis and stochastic opti-
mization , Mrorking Paper, Dept. of Mathema.tics, University of California, Davis, Calif.
[15] Yashin, A., Arjas, E. (1988): A Note on Ra~ldoniIntensities and Conditional Survival Functions. J. Appl. Prob., 2 5 , pp. 630-63.5.